Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{x^2 + 15x + 54}{x^2 - 81}$
First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 15x + 54}{x^2 - 81} = \dfrac{(x + 6)(x + 9)}{(x - 9)(x + 9)} $ Notice that the term $(x + 9)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x + 9)$ gives: $y = \dfrac{x + 6}{x - 9}$ Since we divided by $(x + 9)$, $x \neq -9$. $y = \dfrac{x + 6}{x - 9}; \space x \neq -9$